aa r X i v : . [ a s t r o - ph ] F e b POLLUTION OF SINGLE WHITE DWARFS BY ACCRETIONOF MANY SMALL ASTEROIDS
M. JuraDepartment of Physics and Astronomy and Center for AstrobiologyUniversity of California, Los Angeles CA 90095-1562; [email protected]
ABSTRACT
Extrapolating from the solar system’s asteroid belt, we propose thatexternally-contaminated white dwarfs without an infrared excess may be experi-encing continuous accretion of gas-phase material that ultimately is derived fromthe tidal destruction of multiple small asteroids. If this scenario is correct, thenobservations of metal-polluted white dwarfs may lead to determining the bulkelemental compositions of ensembles of extrasolar minor planets.
Subject headings: planetary systems – white dwarfs
1. INTRODUCTION
An indirect but potentially powerful method to determine the elemental abundances ofextrasolar minor bodies and planets is to measure the photospheric compositions of starsthey pollute. Such studies in main-sequence stars are challenging because the host star’sphotosphere possesses intrinsic metals, making it difficult to determine the amount, if any,of contamination (see Gonzalez 2006). In contrast, heavy elements rapidly settle out ofthe atmospheres of white dwarfs cooler than 20,000 K (Koester & Wilken 2006), so thatphotospheric metals have an external source. We hope to identify which metal-possessingcool white dwarfs have accreted circumstellar rather than interstellar material in order touse these white dwarf atmospheres to measure the bulk elemental composition of extrasolarminor planets.There are strong arguments that the approximately ten single white dwarfs known tohave an infrared excess (Kilic et al. 2006a, 2006b, Kilic & Redfield 2007, Mullally et al.2007, Jura et al. 2007b, von Hippel et al. 2007, Farihi et al. 2008a, b) are accreting tidally-disrupted asteroids. (1) The near-infrared excess discovered around G29-38 (Zuckerman &Becklin 1987) was shown to be produced by dust (Graham et al. 1990) that likely resultedfrom an orbitally-perturbed minor planet (Debes & Sigurdsson 2002) that passed within 2 –the tidal-radius of the host star (Jura 2003). For other white dwarfs, the infrared excess isusually well modeled with a geomerically flat dust disk that lies within the tidal-radius of thestar (Jura et al. 2007b). (2) The infrared spectrum of G29-38 displays strong 10 µ m silicateemission with a spectral shape that resembles dust emission from the zodiacal light ratherthan interstellar silicates (Reach et al. 2005). GD 362, the second white dwarf discoveredto have a near-infrared excess (Becklin et al. 2005, Kilic et al. 2005), also displays a 10 µ msilicate feature characterisitic of asteroidal rather than interstellar dust (Jura et al. 2007a).(3) The white dwarfs with an infrared excess generally show when measured that F ν (24 µ m) << F ν (8 µ m) (Reach et al. 2005, Jura et al. 2007b). There is no evidence for relatively colddust grains outside the tidal-radius (Jura et al. 2007b). (4) The atmospheric composition ofGD 362 is refractory-rich and volatile-poor (Zuckerman et al. 2007), reminiscent of materialin the inner solar system and unlike the interstellar medium. While not as comprehensivelystudied as GD 362, there are at least three other white dwarfs which are makedly deficient incarbon relative to iron as is characteristic of asteroids and very unlike the Sun or interstellarmatter (Jura 2006). (5) Asteroid belts similar the solar system’s can supply the requiredpollution masses (Jura 2006).Only about 1/7 of all contaminated white dwarfs display an infrared excess (Kilic &Redfield 2007); the origin of the pollution of the more numerous contaminated white dwarfswithout an infrared excess is uncertain. Although interstellar accretion could supply themetals in the atmospheres of contaminated white dwarfs (see, for example, Dupuis et al.1993, Koester & Wilken 2006, Zuckerman et al. 2003, Dufour et al. 2007), there aredifficulties with this model. First, at least for helium-rich stars, the accretion of hydrogenrelative to calcium must be suppressed by at least a factor of 100 (Dufour et al. 2007).None of the suggested mechanisms such as magnetic shielding that preferentially excludehydrogen from the star have yet received any observational support (Friedrich, Jordan &Koester 2004). Furthermore, there is no correlation between metal contamination and thestar’s space motion as would be expected from Bondi-Hoyle accretion theory (Zuckermanet al. 2003, Koester & Wilken 2006). Finally, using a very simple model of grain inspiralbecause of Poynting-Robertson drag, Jura et al. (2007b) showed that the predicted fluxesat 24 µ m for Bondi-Hoyle accretion can be more than 100 times greater than the observedupper limits. In view of these difficulties, the alternative of accretion of circumstellar mattermerits consideration.Gaensicke et al. (2006, 2007) have identified two metal-contaminated single white dwarfswith gaseous dust disks that are likely the result of tidal-disruption of a large parent-bodysuch as an asteroid. It is possible that in many cases, the dust orbiting a white dwarf isdestroyed and therefore stars without an infrared excess may still experience accretion fromtidally-disrupted minor bodies. In this paper, we elaborate upon this possibility. 3 –In contrast to hydrogen-rich white dwarfs where the gravitational settling time of metalsis typically less than 100 years (see von Hippel & Thompson 2007), in helium-rich whitedwarfs cooler than 20,000 K, the gravitational settling time may be considerably longer than10 yr (Paquette et al. 1986, Dupuis et al. 1993). Although very uncertain, below, wesuggest that the lifetime of a dusty disk around a white dwarf might be ∼ yr. Therefore,at least for the helium-rich white dwarfs, the absence of an infrared excess might simply bea situation where a single tidally-disrupted asteroid has been fully accreted and the disk hasdissipated but the contamination still lingers in the star’s atmosphere.In § § §
4, wedescribe in somewhat more detail our model of white dwarf contamination by multiple tidal-disruptions of small asteroids. In § § §
2. OVERVIEW
Very little is known about extrasolar asteroid belts. Even the presence of warm dust at afew AU from the star does not tightly constrain the parent body populations (see Wyatt et al.2007). Here, assuming that many other main-sequence stars possess an asteroid belt similarto the solar system’s, we present a single illustrative model to account for the white dwarfdata. There are many unknown parameters; our calculations are aimed to establish thereis at least one plausible scenario where multiple small asteroids can explain the presence ofmetals in white dwarf photospheres. If detailed abundance studies of polluted white dwarfswithout an infrared excess are shown to be consistent with this hypothesis, then a morecomprehensive investigation would be warranted.In our illustrative model, we suggest that the total mass of the asteroid belt when thestar becomes a white dwarf is 10 g, somewhat larger than the solar system’s asteroid beltcurrent mass of 1.8 × g (Binzel, Hanner & Steel 2000). We assume there is a steeplyrising number of lower-mass asteroids, and we then compute that asteroids larger than ∼ ∼ years; during this phasethere is an infrared excess.
3. ASTEROID SURVIVAL DURING THE STAR’S AGB PHASE
Asteroids must survive the AGB phase if they are to pollute the star as a white dwarf.Rybicki & Denis (2001) discuss three processes that can lead to terrestrial planet destructionduring the AGB phase. A planet can induce a tidal bulge in its host star which can theninteract with the planet to drag it inwards. However, asteroids have such small masses, thatthis effect is not important. Below, we consider two other routes for asteroid destruction.
Asteroids may spiral into the host star under the action of gas drag as they orbit throughthe wind of the AGB star. We assume that an asteroid is destroyed if it encounters its ownmass in the wind. This destruction occurs because of hydrodynamic friction rather thansputtering which is negligible since the colliding gas atoms have kinetic energies of only afew eV (see Tielens et al. 1994). At distance D from the star in a spherically symmetricwind of mass loss rate ˙ M ∗ with outflow speed, V wind , the gas density, ρ wind , is ρ wind = ˙ M ∗ πD V wind (1) 5 –As long as gravitational focusing is negligible, the rate of encountering mass, ˙ M enc , by theasteroid is: ˙ M enc = π R ast ρ wind V ast (2)where the asteroid of radius R ast has orbital speed V ast . For this evaluation, we assume thatthe asteroid’s radius and mass are constant during the AGB phase. For circular motion: V ast = (cid:18) G M ∗ D (cid:19) / (3)If the time scale for mass loss is slow compared to the asteroid’s orbital period, thenthe specific angular momentum of the asteroid is approximately conserved and: D M ∗ = D i M i (4)for an initial distance, D i , from a star of initial mass, M i . As the star loses mass and M ∗ diminishes, then the asteroid’s orbit expands. Combining equations (1)-(4), then during thestar’s AGB phase, the total encountered mass, M enc is: M enc = Z ˙ M enc dt ≈ R ast G / M / i V wind D / i (5)In arriving at the last expression in equation (5), a simplified and slightly altered version ofequation (3) of Duncan & Lissauer (1998), we assume M i >> M f where M f is the final ofthe star (see, for example, Weidemann 2000).If ρ ast is the density of the asteroid whose mass is (4 πρ ast R ast ) /
3, then equation (5)leads to the condition that the minimum radius of a surviving asteroid, R min , is: R min = 3 M / i G / π D / i ρ ast V wind (6)We only consider asteroids with initial distances from the host star greater than 2 AUbecause interior to this region, there may be a quasi-static region around the AGB star witha density much greater than given by equation (1) (Reid & Menten 1997). Beyond 2AU,we assume an outflow velocity of 2 km s − (Keady, Hall & Ridgway 1988), a speed that isappreciably smaller than the typical wind terminal speed of 15 km s − (Zuckerman 1980).We adopt ρ ast = 2.1 g cm − , as inferred for Ceres (Michalak 2000).We show in Figure 1 the results for R min as a function of D i for initial stellar massesof 1.5 M ⊙ , 3 M ⊙ and 5 M ⊙ . As can be seen from equation (6), the value of R min diminishes 6 –rapidly as D i increases since less mass is encountered in the wind. Also, R min increases withthe intial mass of the star since the star ejects more mass for the asteroid to encounter.Depending upon the initial distance from the host star and its mass, we find that asteroidslarger than 1-10 km in radius are likely to survive the drag induced by the AGB wind. Subjected to the star’s high AGB luminosity, an asteroid might thermally sublimate,and we now estimate the minimum size asteroid that survives this process. If the star’s highluminosity phase has duration, t AGB , then, the minimum size of a surviving asteroid is: R min = dR ast dt t AGB (7) dR ast /dt is the rate at which an asteroid’s radius shrinks because of sublimation: dR ast dt = ˙ σ ( T ) ρ ast (8)where ˙ σ ( T ) denotes the mass production rate from the dust per unit area (g cm − s − ) as afunction of the asteroid temperature, T . Assuming pure olivine, then:˙ σ ( T ) = ˙ σ r T T e − T /T (9)where, converted to cgs units, ˙ σ = 1.5 × g cm − s − and T = 65,300 K (Kimura et al.2002). Different materials may have different sublimation rates so our results are sensitive tothe presumed asteroid composition. We adopt olivine as this appears to be the most likelycarrier of the 10 µ m silicate emission seen in G29-38 (Reach et al. 2005) and GD 362 (Juraet al. 2007a) as well as being common in the solar system. Assuming that the asteroids havea negligible albedo, we write that: T = (cid:18) L ∗ π σ SB D (cid:19) / (10)where L is the luminosity of the star and σ SB is the Stephan-Boltzmann constant. While onthe AGB, we assume the star has a luminosity near 10 L ⊙ or larger for t AGB = 2 × yr(Jura & Kleinmann 1992).Using equation (8)-(10), we show in Figure 2 the minimum size asteroid from equation(7) for stellar luminosities of 1 × L ⊙ and 2 × L ⊙ as a function of the asteroid’sdistance from the star. We neglect the orbital drift of the asteroid discussed above. Because 7 –the thermal sublimation rate is temperature sensitive, asteroid survival decreases rapidly asthe star’s luminosity increases or for asteroids that orbit relatively close to the star. We seefrom Figure 2 that depending upon the luminosity of the star, asteroids with radii between1 and 10 km are likely to survive if they orbit beyond 3 or 4 AU.
4. ACCRETION OF MANY SMALL ASTEROIDS
Eventually, those asteroids that survive a star’s AGB evolution, may have their orbitsperturbed so that they come within the tidal radius of the white dwarf (Duncan & Lissauer1998, Debes & Sigurdsson 2002). When this occurs, the asteroid is shredded into dust. Ifthe asteroid is relatively small and if there is a pre-existing disk from destruction of previousasteroids, then the shredded debris of the new asteroid is destroyed by sputtering in thepre-existing disk, and the circumstellar matter is largely gaseous. If, instead, the asteroid ismore massive than the pre-existing disk, there is not enough gas to sputter the debris fromthe asteroid, and a disk with a large amount of dust at the inclination angle of the largeasteroid is created. This more massive disk subsumes the low-mass pre-existing gaseous disk.Eventually, however, this large disk is accreted onto the star and the system relaxes to itssteady state configuration of a largely gaseous disk.Denote the total asteroid belt total mass as M belt ( t ), where t denotes the cooling time ofthe white dwarf. Let t orbit be the mean lifetime of an asteroid before it is perturbed into thetidal radius of the star. Assuming that all of the shredded asteroid ultimately is accreted,then the average metal-accretion rate, < ˙ M metal > , is: < ˙ M metal > = M belt ( t ) t orbit (11)If the circumstellar gaseous disk around a white dwarf has a lifetime, t gas , then the massof a typical disk, M disk , is: M disk = < ˙ M metal > t gas (12)We propose that if an asteroid has a mass larger than M disk , then a single event dominates thecircumstellar environment, and the star exhibits an infrared excess. If, however, an asteroidarrives with a mass lower than M disk , then this newly-shredded asteroid’s dust debris israpidly vaporized and the disk remains gaseous.We picture that all polluted white dwarfs possess in a steady state a gaseous disk,and, occasionally, a massive dusty disk. It is likely that a white dwarf takes relatively littletime to enter its “normal” steady state with a gaseous disk. The first tidally-disruptedasteroid is shredded into dust which can persist for a long time since there is not likely to be 8 –much viscosity in the dust particles. The second tidally-disrupted asteroid also is shreddedinto dust, but its orbital inclination is likely to be different from that of the first asteroid.Consequently, there will be grain-grain collisions approaching 1000 km s − within the tidalradius. These collisions are very destructive and large amounts of gas will be released.Eventually, as described below, the disk will evolve into a largely gaseous system. t gas To progress further, we now estimate the lifetime of a white dwarf’s circumstellar diskwhich we presume is controlled by viscous dissipation. We could extrapolate from cataclysmicvariables to single white dwarfs. However, disks derive from destroyed asteroids probablyhave little hydrogen, and their compositions are very different from that of the disks incataclysmic variables. For a gaseous circumstellar disk, the time for viscous dissipation is(King, Pringle & Livio 2007) t gas ∼ D V ast αV th (13)where α is the usual dimensionless parameter, and V th is ∼ k B T /m ast where k B is the Boltz-mann constant, T the temperature and m ast the mean atomic weight of the asteroidal matter.The most uncertain factor is α . As discussed by King et al. (2007), empirical estimates of α for thin, fully ionized disks are between 0.1 and 0.4, but theoretical simulations yield valuesof α that are at least an order of magnitude lower. In FU Ori stars, values of α as low as0.001 are inferred. The viscosity also may be a function of the gas ionization and decrease asthe gas becomes more neutral. Here, because the disks around single white dwarfs may belong-lived, we assume that α is 0.001, but this value is extremely uncertain. We use modelsof the infrared emission from disks (Jura et al. 2007b) to adopt a characteristic disk size, D , of 3 × cm. At this distance from the star, V ast = 5 × cm s − , and at least fordusty disks, a representative temperature is 300 K (Jura 2003). If m ast equals that of silicon,then t gas ≈ × yr. Although this estimate of the time scale is very uncertain, it is longcompared to the time between arrivals of small asteroids within the tidal zone. We now show that if a relatively small asteroid impacts a gaseous disk that the resultinginfrared excess from dust debris has a short duration compared to the characteristic disklifetime. In the solar system, asteroids in the main belt have a mean inclination, i , of 7.9 ◦ (Binzel et al. 2000), and in white dwarf systems, it is unlikely that a newly tidally-disrupted 9 –asteroid has exactly the same orbital inclination as the pre-existing gaseous disk. As a result,as the debris from the asteroid orbits through the disk, it is eroded by sputtering.Let Σ disk (g cm − ) denote the mean disk surface density:Σ disk = M disk πD (14)If spherical grains of radius a move with speed v through a disk with number density, n ,then the mass loss from sputtering is: ddt (cid:18) πρ ast a m ast (cid:19) = − y (cid:0) π a n v (cid:1) (15)where y is the sputtering yield. If the grains and the disk have the same composition sincethey are both derived from asteroids, then each passsage through the disk erodes a layer ofthickness, ∆ a pass , where: ∆ a pass = y Σ disk ρ ast sin i (16)With two passages per orbit, then in time t with t>> t orbit , the minimum radius of a grainthat survives, a min is: a min = 2∆ a pass tt orbit (17)The infrared excess from a disk depends upon its optical depth which we now showrapidly becomes optically thin. Let χ denote the opacity of the dust. Assuming that thegrain optical cross section is given by its geometric cross section and making the conservativeassumption to assume that all the dust particles have size a min , then: χ = 34 ρ ast a min (18)If the mass of the asteroid is less than the mass of the disk, then the vertical optical depththrough the disk, τ z , is bounded so that: τ z ≤ χ Σ disk (19)Using equations (16) - (19), then: τ z ≤ i y t orbit t (20)Light from the star of radius R ∗ , illuminates the disk at a slant angle of R ∗ /D , and theoptical depth in the disk, τ disk , to light at this angle is: τ disk ≈ τ z DR ∗ = 3 D sin i R ∗ y t orbit t (21) 10 –From equation (21), τ disk diminishes with time as grains are destroyed. We now assumethat the typical atom in the disk is silicon. The orbital speed at the disk is near 500 km s − and with an inclination angle of 7.9 ◦ , the characteristic speed and kinetic energy of a siliconatom impacting a grain are ∼
70 km s − and ∼
700 eV, respectively. Using the models ofTielens et al. (1994), we adopt y ≈ D/R ∗ = 30, as representative of a disk withinthe tidal zone of the star, equation (21) shows that τ dist becomes less than unity within ∼ ∼
10 years, this implies thatthere is appreciable infrared emission from the system for at most ∼
600 years. The trueduration of an excess is probably much shorter since, for example, most incoming asteroidshave masses appreciably smaller than that of the disk so that the bound given in equation(19) is conservative. Also, we have probably substantially overestimated the opacity of thedust debris by assuming all the particles have the minimum size.
We propose that when a massive asteroid’s orbit is perturbed so it is tidally destroyed,then a dust disk results. If analogous to Saturn’s rings, this might only take ∼
100 orbitaltimes or ∼ yr. However, we anticipate that the dust disk is bombarded by orbitally-perturbed small asteroids and the resulting grain-grain collisions produce gas in the disk.For a mass distribution of asteroids given by equation (23) below, the median mass of a largeasteroid that creates a dust disk is 2 M disk . A dust disk derived from the median-sized largeasteroid accumulates by further collisions of small asteroids, its own mass in gas in a timescale of 2 t gas . Furthermore, the asteroid may have had a substantial amount of internalwater which can be released into the gas phase as the asteroid is destroyed. Estimates forthe percentage by mass of internal water in Ceres range between 0.17 and 0.27 (McCord &Sotin 2005, Thomas et al. 2005). If the viscous time scales inversely as the fraction of matterthat is gaseous, then in the absence of adding any additional gas from small asteroids, t dust would be between a factor of 4 and 6 longer than t gas . Although extemely uncertain, weadopt on average that t dust ∼ t gas or perhaps 1.5 × yr. 11 –
5. MODEL ASSESSMENT
We now evaluate the model that asteroids are the main source of contamination manyexternally-polluted white dwarfs. In our “standard model” the mass of asteroid belt, M belt ( t )is taken to equal 10 g at the onset of the white dwarf phase when t = 0. We assume thatthere is a characteristic time, t orbit , to perturb an asteroid into a tidally-disrupted orbit.Therefore: M belt = M belt (0) e − t/t orbit (22)This expression for an exponential decay of the asteroid belt is at best a rough approximation.Wyatt et al. (2007) discuss models where the mass varies as t − because of mutual collisionswhile including orbital perturbations, more complex scaling models for the exponential decaymay apply (Dobrovolskis, Alvarellos & Lissauer 2007).A critical parameter in our analysis is t orbit . To-date, simulations for the evolution ofasteroid belts around white dwarfs have been somewhat limited, although Duncan & Lissauer(1998) and Debes & Sigurdsson (2002) computed various scenarios. For the future evolutionof the solar system, Duncan & Lissauer (1998) followed the orbits of 5 asteroids, and foundthat 1 was always unstable. In two out of their four simulations, a second asteroid, Pallas,became unstable in 5 × yr. To reproduce the data, we adopt values of t orbit near 1 Gyr.However, a more simulations of for asteroid belt evolution during the white dwarf phase areneeded.We assume that the asteroid belt has a mass distribution, n ( M ) dM , given by a powerlaw so that: n ( M ) dM = A M − dM (23)where the constant A is derived by the normalization condition. This power law variation of M − is slightly steeper than the “classical” estimate for the solar system that n ( M ) variesas M − . (O’Brien & Greenberg 2005), but it is computationally convenient and adequatefor systems about which we know very little. If M belt is the total mass of the asteroid beltwith minimum mass, M min and maximum mass M max , then A = M belt (cid:18) ln (cid:20) M max M min (cid:21)(cid:19) − (24)As in the solar system, we assume the largest asteroid has an appreciable fraction of thetotal mass of the entire system, and we adopt M max ( t ) = 0.2 M belt ( t ) Thus, initially, thelargest asteroid has a mass of 2 × g, somewhat larger than the mass of Ceres which is9.4 × g (Michalak 2000). We assume that M min is independent of time and equals 2 × g, the mass of an asteroid with radius 3 km and density 2.1 g cm − . In this model, the 12 –total number of asteroids initially is 3 × . With t orbit = 1 Gyr, this would result in anasteroid being tidally disrupted every 300 yr. In the context of our model, we now compute which white dwarfs may have an infraredexcess. This requires estimating < ˙ M metal > which we take as: < ˙ M metal > = M t orbit e − t/torbit (25)To compare this quantity with observables, we estimate the white dwarf’s cooling time fromits effective temperature. We interpolate from the calculations of Winget et al. (1987) andHansen (2004) and take t = 0 . (cid:18) T eff (cid:19) (26)where t denotes the cooling time (Gyr). This approach neglects the variation in stellar radiiand masses. If a white dwarf exhibits an accretion rate above < ˙ M metal > , then, accordingto our model, it is experiencing accretion from a single large asteroid and should possess aninfrared excess. We show in Figure 3 a plot of accretion rate vs. stellar effective temperaturefor externally-polluted hydrogen-rich white dwarfs where the accretion rate can be reliablyestimated. In this plot, we also show the value of < ˙ M metal > for M = 10 g and t orbit varying from 0.5 Gyr to 2.0 Gyr. The model and the data are consistent for most but notall stars. The adopted estimated mass in the asteroid belts of 10 g is larger than in thesolar system; this point is discussed below in § M metal = 4 × g s − and T eff = 6600 K (Koester & Wilken 2006). Perhaps this star has a particularly long-lived andmassive asteroid belt. Any model to explain the contamination of white dwarfs should account for the relativenumbers of stars with and without an infrared excess. We hypothesize in § N ( M > M disk ), is: N ( M > M disk ) ≈ AM disk (27)The total rate at which these asteroids are perturbed into orbits where they are tidally-destroyed is N ( M > M disk ) /t orbit . Therefore, if t dust is the duration of a dust disk, thefraction of time, f IR , that the white dwarf exhibits an infrared excess is: f IR = N ( M > M disk ) t dust t orbit (28)Using equations from above, this expression becomes: f IR = (cid:18) t dust t gas (cid:19) (cid:18) ln M max M min (cid:19) − (29)Evaluation of expression (29) is insensitive to the exact values of M max and M min . From ourestimates given above, we find that f IR = 0.062 ( t dust /t disk ). In § .
3, we argued that t dust ∼ t disk . Although extremely uncertain, the prediction from equation (29) is therefore that f IR ∼ One motivation for studying the composition of contaminated white dwarfs is that itenables the measurement of the bulk elemental composition of extrasolar minor planets. Forexample, the photosphere of the white dwarf GD 362 was found to be markedly deficientin volatiles such as carbon and sodium relative to refractories such as calcium and iron(Zuckerman et al. 2007), a pattern reminiscent of the inner solar system where the Earthand asteroids also are very volatile deficient.Wolff et al. (2002) reported relative abundances of iron, magnesium, silicon, calciumand carbon in 10 helium-rich white dwarf. Although in many cases the uncertainties are verylarge, three of these stars exhibit a carbon abundance less than that of iron, and thereforea value of n (C)/ n (Fe) that is at least a factor of 10 smaller than solar. These three starshave a composition similar to that of the Earth or chondrites; they do not appear to haveaccreted interstellar matter (Jura 2006). One of these three stars, GD 40, also displaysan infrared excess (Jura et al. 2007b), consistent with the hypothesis that it is accretingchondritic material. However, the other two stars do not display an infrared excess (Farihiet al. 2008b) which, however, as discussed below, does not preclude the asteroid accretionscenario. 14 –Desharnais et al. (2008) report FUSE data for 5 helium-rich white dwarfs. Of these 5stars, GD 378 has n (C)/ n (Fe) = 2.5, within the errors not grossly different from the solarvalue of 8.3 (Lodders 2003). There are mostly only upper limits to the abundances forGD 233, a second star in this sample. However, G270-124, GD 61 and GD 408 all havevalues of n (C)/ n (Fe) less than 0.3. These three stars are therefore candidates for havingaccreted chondritic material. G270-124 does not display an infrared excess shortward of 8 µ m (Mullally et al. 2007). As far as we know, GD 61 has not observed for an infrared excess.There is a hint of an excess at 7.9 µ m for GD 408 (Mullally et al. 2007), but the signal tonoise was not good enough for a definitive conclusion.Not all contaminated white dwarfs appear to have accreted chondritic material. G238-44 has been found to display n (C)/ n (Fe) near 2.5 (Dupuis et al. 2007); this star does notdisplay any infrared excess (Mullally et al. 2007). As with GD 378, the source of the externalmatter is unknown.There are at least two possible explanations for the absence of a near infrared excessaround externally-polluted helium-rich white dwarfs. First, as discussed above, the cir-cumstellar disk could be largely gaseous. Second, the disk may have dissipated before thecontaminants have settled out of the star’s atmosphere. The settling time of heavy metalsin the atmospheres of these stars with effective temperatures between 15,000 K and 21,000K ranges from ∼ × yr to ∼ × yr (Dupuis et al. 1993), comparable to or longerthan our very rough estimate of the duration of a dust disk of 1.5 × yr. Above, we have adopted the view that a disk is either essentially all dust or all gas.However, it is possible that some dust survives in disks that are bombarded by numeroussmall asteroids. Consider the case where the interior of the disk is gaseous yet the outerportion remains dusty, and thus the disk distribution may have an especially large centralhole. All but one of the ten white dwarfs currently known to have an infrared excess has dustas warm as 1000 K (see Jura et al. 2007b). This innermost temperature may be determinedby the location where grain sublimation becomes rapid. However, G166-58 (also known asLHS 3007 or WD 1456+298), a metal polluted white dwarf with a metal accretion rate of 3 × g s − (Koester & Wilken 2006), was found by Farihi et al. (2008a) to have an excessat 5.7 µ m and 7.9 µ m, but not at shorter wavelengths. Following Jura et al. (2007b), wecompute a model with a flat disk to reproduce the data. We adopt a stellar temperature of7390 K and a ratio of the star’s radius (9.1 × cm) to the star’s distance from the Sun( ≈
30 pc), R ∗ /D , of 1.0 × − . We fit the infrared data with an inner disk temperature of 15 –400 K (corresponding to a physical distance from the star of 30 R ∗ , Jura [2003]), an outerdisk temperature of 300 K (corresponding to a physical distance from the star of 43 R ∗ ),and cos i = 0.4. This model disk lies well within the tidal zone which extends to ∼ R ∗ (Davidsson 1999). The comparison between observations and the model is shown in Figure4; the fit is good enough that this model is a serious contender for explaining the data. Thus,this system may be in an intermediate phase of its disk evolution where the dust disk hasan usually large hole because bombardment destroyed dust in the inner disk. Gaensicke et al. (2006, 2007) report emission lines from two disks; most polluted whitedwarfs do not display emission lines even though we postulate the presence of a circumstellarreservoir from which they accrete. We suggest that the systems with detected emission areunusual because the gas has an unusually high excitation temperature, T ex . If the star haseffective temperature, T ∗ , and subtends solid angle, Ω ∗ , while the disk subtends solid angle,Ω disk , then a line is seen in emission if: B ν ( T ex ) Ω disk ∼ B ν ( T ∗ ) Ω ∗ (30)For SDSS 1228+1040, the analysis of Gaensicke et al. (2006) yields Ω disk / Ω ∗ = 1.2 × .Since T ∗ = 22020 K, then for the calcium lines near 8500 ˚A, the above criterion requires T ex >
6. DISCUSSION
There are a number of similarities between the asteroid belt in the solar system and theproposed asteroid belts around white dwarfs. We invoke asteroid belt masses of 10 g atthe time the star enters its white dwarf phase, this mass is about a factor of 6 larger thanthe mass of the asteroid belt in the solar system. However, if stars form with a Salpetermass function, n ( M ) dM , that varies as M − . and if the lowest mass star that currently is 16 –dying has a mass of 0.9 M ⊙ , then the median main-sequence mass of a dying star is ∼ ⊙ . There may be a linear scaling between the mass of a star’s asteroid belt and its main-sequence mass. Furthermore, 1.5 M ⊙ stars spend only ∼ ⊙ star becomes a white dwarf, a simple linear extrapolation from thesolar system suggests that it would possess 4 × g, and thus approximately resemble thesolar system.A minimum mass of an extrasolar asteroid can be estimated from the mass of contam-inants in the atmosphere of a contaminated helium-rich white dwarf. Since calculations forthe mass in the outer mixing zone differ by as much as a factor of 10 (Dupuis et al. 1993,MacDonald et al. 1998), these estimates are uncertain. At the moment, the most pollutedknown He-rich white dwarf is HS 2253+8023 (Wolff et al. 2002), and the inferred mass ofits accreted asteroid is ∼ g (Jura 2006), comparable to the mass of Ceres.The relative numbers of polluted white dwarfs with and without an infrared excess isexplained by a mass distribution of extrasolar asteroids that varies as M − , approximatelywhat is observed in the solar system. Thus extrasolar asteroid belts may possess a sizedistribution similar to that of asteroids in the solar system.Remarkably, in marked contrast to the Sun, carbon appears to be less abundant bynumber than iron or silicon in at least a few extrasolar asteroids. Similarly, as measuredfrom chondrites (Lodders 2003), asteroids in the solar system have a similarly low carbon toiron abundance ratio. As argued previously (Jura 2006), determinations of the abundancesin the atmospheres of white dwarfs can serve both to evaluate the model of interstellaraccretion and, at least in some cases, to measure the composition of extrasolar asteroids.
7. CONCLUSIONS
We suggest that the contamination of many externally-polluted white dwarfs withoutan infrared excess results from tidal-disruptions of numerous small asteroids. This scenariois consistent with the hypothesis that the masses, size distributions and compositions ofextrasolar asteroids are similar to those of the solar system’s asteroids.This work has been partly supported by NASA. 17 –
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This preprint was prepared with the AAS L A TEX macros v5.2.
20 –Fig. 1.— Minimum surviving asteroid radius vs. initial distance from the star for stellarwind drag from equation (7). The solid, dotted and dashed lines are plotted for stars ofinitial masses of 1.5 M ⊙ , 3.0 M ⊙ and 5.0 M ⊙ , respectively. 21 –Fig. 2.— Minimum surviving asteroid radius vs. distance from the star for olivine sublima-tion ( § × L ⊙ and 2.0 × L ⊙ , respectively. 22 –Fig. 3.— Metal accretion rates vs. stellar effective temperature. The filled circles representstars with an infrared excess while the open triangles represent stars without an infraredexcess. The data are taken from Farihi et al. (2008a) and Jura et al. (2007b); the metalaccretion rates are taken from Koester & Wilken (2006) and scaled according to the procedurein Jura et al. (2007b). The lines represent < ˙ M metal > as a function of the star’s effectivetemperature from § g and the dotted,solid and dashed lines represent values of t depl of 0.5 Gyr, 1.0 Gyr, and 2.0 Gyr, respectively.The solid circles are expected to lie above the curves while the open triangles are predictedto lie below the curves. 23 –Fig. 4.— Comparison of data showing 1 σσ